stats report

Infectious Disease Dynamics with Immunization

Age Structure

The final thing that we will do with epidemic models is to study the effect of immunization. In order to do this, we will need a model that has age structure. Our previous models do not track age. They treat everyone as identical in the population. Howeer, the dynamics of vaccination are highly dependent on age structure.

We can write differential equation models that include age structure. However, this would require (partial differential equations). This a more advanced area that we won’t study in this class. Instead, we will modify the simulation model.

Previously, the population was represented with a vector pop that stored the disease status (susceptible, infected, recovered) for each member of the population. Now, we will represent the population with a matrix, that stores disease status, age, age at infection, and sex:

The function below initializes the population: #function that generates poisson(lambda) values until it gets a non-zero value trunc_pois<-function(lambda) { d=0

while(d==0) d=rpois(1,lambda) return(d)

}

FindWorkPlace<-function() {

}

#Make random households #F0 fanilies with kids. All have two parents, make and fenale #S0 households adults only of random size

make_households<-function(pop,H0) { for(f in 1:F0)

{ ageM=runif(20,50) #qge of mother ageD=ageM+rnorm(0,5) #age of father nKids=trunc_pois(1)

mom=c(0,ageM,NA,1,f,)

}

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}

initialize_pop<-function(S0,I0,maxAge) {#initialize the population. pop0 is matrix with pop0[i,1]=susceptble/infected/recovered status

#pop0[i,2] is current age. pop0[i,3] will be assigned age at infection, pop0[i,4]=1 if female

pop0=matrix(nrow=S0+I0,ncol=4)

#assigned susceptible/infected status: pop0[1:S0,1]=0 pop0[(S0+1):(S0+I0),1]=1 pop0[(S0+1):(S0+I0),3]=0 #assign age at infection

#assign ages. all ages have equal probability. pop0[,2]=sample(maxAge*365,S0+I0,replace=T)

#assign sex (only females reproduce) pop0[,4]=ifelse(runif(nrow(pop0))<=0.5,1,0) #1 if female, 0 if male

return(pop0)

}

Now, we can have important parameters (infection rates, death rates, etc) vary by age. The functions nelow define age-specific rates. Each function takes age a as input and outputs the appropriate rate. Note that these currently are not very realistic. I didn’t use any data in setting the rates and instead just took very simple values. However, we can easily incorporate detailed data. InfectionRatebyAge<-function(a) { #function returns the infection rate for an individual of age a

# age is in days

if(a<=5*365) { #less than 5 years old return(0.015) } else if(a<=10*365) { #5-10 years old

return(0.012) } else if(a<=69*365) {#10-69 years old

return(0.01) } else{ #> 69

return(0.02) }

}

RecoveryRatebyAge<-function(a) { #function returns the infection recovery rate for an individual of age a

# age is in days

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if(a<=5*365) { return(0.1)

} else if(a<=10*365) { return(0.1)

} else if(a<=69*365) { return(0.1)

} else{ return(0.1)

}

}

BirthRate<-function(a) {#function returns the birth rate for an individual of age a

# age is in days #currently set so that women from 18 to 45 have same fertility rate #if less than 18 or more than 45, fertility is zero. #this is not very realistic. We can improve with data if(a<=18*365){

return(0) } else if(a<=45*365) {

return(1/27) } else {

return(0) }

}

BackgroundMortalitybyAge<-function(a) { #function returns the infection rate for an individual of age a

if(a<=5*365) { return(0.015/365)

} else if(a<=10*365) { return(0.015/365)

} else if(a<=69*365) { return(0.015)

} else{ return(0.05/365)

}

}

The functions below do all do the updating of the population. The functions infect, recover, backgroundmortality, and birth work similarly to how they did previously. However, they are now modified to account for age structure. The functions aging, clear_dead and cap_pop are new: aging ages the population by one day every day andclear_dead removes dead people from the population each day. The function cap_pop puts a cap on the population. If the population exceeds maxpop at the end of the day, then the function randomly selects people and removes them from the population. This function is required to prevent the population from growing exponetially and making the program run very slowly. Ideally, we would have death rates and birh rates balanced so that the population stays constant. However, this is difficult to acheive in practice and so we force a cap on the population.

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aging<-function(pop) {#function to age everyone by one day

pop[,2]=pop[,2]+1

}

#Function to make new infections: #"flips coin" for each population individual to see whether they get the disease. #if so, their status is changed from susceptible (0) to infected (1). infect<-function(pop,InfectRate) { nI<-sum(pop[,1]==1) #number of infecteds

nS<-sum(pop[,1]==0) #number of susceptibles pop1=pop #create copy of pop

susceptibles=which(pop[,1]==0) #make vector giving positions of suscepibles

beta=sapply(pop[susceptibles,2],InfectRate) #sapply applies the function to the vector #produces a vector of infection probabilities. #where beta[i] is infection prob for ith susceptible

pi=1-(1-beta)^nI #probility of susceptible being infected. Calcs seperate prob for each individual, based on # that individuals age-specific infection prob.

#pi=ifelse(beta*nI<=1,beta*nI,1)

pop1[susceptibles,1]=ifelse(runif(nS)<pi,1,0) #check if each susceptible becomes infected

return(pop1) }

#function to make individuals recover #"flips coin" for each infected individual to see whether they recover from the disease. #if so, their status is changed from infected (1) to recovered. recover<-function(pop,RecRate) {

nI<-sum(pop[,1]==1) #number of infected pop1=pop

infecteds=which(pop[,1]==1)

gamma=sapply(pop[infecteds,2],RecRate) #sapply applies the function to the vector #produces a vector of infection probabilities.

pop1[infecteds,1]=ifelse(runif(nI)<gamma,2,1) #check if each infected recovers. This is the "coin flip"

return(pop1)

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}

#function to check whether each individual dies from non-disease causes #MortRate is a function that gives the mortality rate for each age backgroundmortality<-function(pop,MortRate) {

pop1=pop n=nrow(pop1) #current population size

#note that this doesn't distinguish between living and dead. That is, dead people can die again. This does not #affect anything.

nu=sapply(pop[,2],MortRate) #produces a vector of mortality rates for everyone in the population #This is speficic to their age

pop1[,1]=ifelse(runif(n)<=nu,3,pop1) #check if each person dies . If so, write 3, if not write current value

return(pop1)

}

#function to check whether each individual dies from disease diseasemortality<-function(pop,m) {

nI<-sum(pop==1) #number of infected pop1=pop pop1[pop==1]=ifelse(runif(nI)<=m,3,1) #check if each infected dies

return(pop1) }

birth<-function(pop,birthrate) { #function to check for births

#birthrate is a function that gives fertility rate by age.

females<-pop[pop[,4]==1,] #find the females nf<-nrow(females) #number of females

nbaby<-sum(runif(nf)<sapply(females[,2],birthrate)) #number of babies born today

if(nbaby>0) #if any babies are born today, then add them to the population { babies<-matrix(nrow=nbaby,ncol=4)

babies[,1]=0 #babies born susceptible (ignoring possibility of temporary immunity from mother) babies[,2]=0 #age 0 babies[,4]=ifelse(runif(nbaby)<=0.5,1,0) #assign sex return(rbind(pop,babies))

}

return(pop) #if no babies, then return pop as is

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}

clear_dead<-function(pop) {#function to remove dead from population

return(pop[-which(pop[,1]==3),]) }

cap_pop<-function(pop,maxpop) {#this function caps the population. If the population size exceeds maxpop, it randomly selects (n-maxpop) individuals # and removes them from the population

n<-nrow(pop)

if(n<=maxpop) return(pop)

remove<-sample(1:n,(n-maxpop),replace=F)

return(pop[-remove,])

}

Finally, here is the main program. This follows the same format as the model without age structure, but with functions that account for age structure. #initialize the population pop=initialize_pop(S0,I0,max_age)

#initialize the output variables: St=S0 It=I0 Rt=0 Mt=0 At=NULL Vt=0 Qt=NULL RI=NULL #will store proportion of

#this is the main part of the program. This loops over time. Each time step is one day. On each day, it checks whether each #person has caught the disease, recovered from the disease, died from non-disease causes, died from disease, or had a baby. for (time in 1:endtime) { if(time/(10*365)==floor(time/(10*365))) print(time/365) #print time every ten years to track progress

pop=infect(pop,InfectionRatebyAge) pop=recover(pop,RecoveryRatebyAge) pop=backgroundmortality(pop,BackgroundMortalitybyAge) pop=diseasemortality(pop,m) pop=birth(pop,BirthRate) pop=aging(pop)

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St=c(St,sum(pop[,1]==0)) It=c(It,sum(pop[,1]==1)) Rt=c(Rt,sum(pop[,1]==2)) Mt=c(Mt,sum(pop[,1]==3)) At=c(At,mean(pop[,3],na.rm=T)) Vt=c(Vt,sum(pop[,5]==1))

#Qt=c(Qt,St*mean(sapply(pop[,2],InfectionRatebyAge))/mean(sapply(pop[,2],RecoveryRatebyAge)))

pop=clear_dead(pop) pop=cap_pop(pop,maxpop)

if(time/(10*365)==floor(time/(10*365))) if(sum(pop[,1]==1)<1) pop[1,1]=1 #if infected drops below 1 add 1 every ten years.

}

Here is a run of the model with age structure. We see that it goes through large amplitude cycles, with disease outbreaks about once every 20 years. In each outbreak, most of the population is infected (we can tell this because the number of susceptibles gets close to zero). The cycles are irregular both in amplitude and period, suggesting the possibility of chaotic dyanmics (we will discuss chaos in the context of population dynamics, our next topic). They don’t show any sign up damping over the 1000 years plotted (in fact, I ran it for 2000 years total and there was not sign of damping over that range either). I am not sure why the cycles are so much bigger with the age stuctured population.

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0 200 400 600 800 1000

0 5

0 0

1 5

0 0

2 5

0 0

time (years)

n u

m b

e r

Susceptible Infected Recovered

Immunization

Next, we will consider the effect of immunization.

From Wikipedia:

A vaccine is a biological preparation that provides active acquired immunity to a particular disease. A vaccine typically contains an agent that resembles a disease-causing microorganism and is often made from weakened or killed forms of the microbe, its toxins, or one of its surface proteins. The agent stimulates the body’s immune system to recognize the agent as a threat, destroy it, and to further recognize and destroy any of the microorganisms associated with that agent that it may encounter in the future. Vaccines can be prophylactic (example: to prevent or ameliorate the effects of a future infection by a natural or “wild” pathogen), or therapeutic (e.g., vaccines against cancer are being investigated).

The administration of vaccines is called vaccination. Vaccination is the most effective method of preventing infectious diseases; widespread immunity due to vaccination is largely responsible for the worldwide eradication of smallpox and the restriction of diseases such as polio, measles, and tetanus from much of the world. The effectiveness of vaccination has been widely studied and verified; for example, the influenza vaccine, the HPV vaccine, and the chicken pox vaccine. The World Health Organization (WHO) reports that licensed vaccines are currently available for twenty-five different preventable infections.

The widespread adoption of vaccines has greatly reduced the incidence of many diseases. Smallpox, formerly one of the largest killers of humans, has been completely eradicated from the world (there are samples at two locations in the world: the CDC in Atlanta, and a government lab in Russia).

In the case of measles, there were an estimated 2-3 million deaths world wide each year before the vaccine was introduced. That number has dropped to under 100,000 per year (e.g. an estimated 90,000 measles deaths in

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2016). These deaths are predominantly in developing countries where there are not widespread immunization programs.

In the US, there were 700,000 measles cases in the US in 1958 (before the adoption of the measles vaccine), resulting in 558 deaths. Since the vaccine came into wide use, the number of cases dropped to under 100 per year (Figure 1).

The table in Figure 2 summarizes measles cases in 2008:

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reference:https://www.cdc.gov/mmwr/preview/mmwrhtml/mm5718a5.htm

We see that most cases were from unvaccinated cases. Although measles is very rare in the US, it is still fairly common in some developing countries. All or nearly all outbreaks in the US start with people traveling from other countries.

Figure 4 below shows measles in recent years (source=CDC). Unfortunately, measles has been on the increase because of the unfortunate trend of parents not vaccinating their children.

Vaccine Schedule

In order to maximize protection, it is recommended that children be vaccinated as soon their immune system is sufficiently developed to respond to a particular vaccine. This has led to a complicated schedule of recommended ages for vaccinations. The link below shows the schedule for the US.

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https://www.cdc.gov/vaccines/schedules/hcp/imz/child-adolescent.html

Children in the US typically receive the vaccines during their regularly checkups at the pediatrician. The pediatrician cannot force children to get vaccines. However, all 50 states have laws requiring children to be vaccinated before entering kindergarten in public schools.

Unfortunately, there has been a trend towards parents not vaccinating their children in the US and other western countries. This is based on 1) mistaken beliefs about the safety of vaccines and 2) people not under- standing how devastating these diseases once were for children before vaccines were introduced. Fortunately, most parents still get their children vaccinated and these diseases are largely kept in check. However, there have been outbreaks of measles, whooping cough, and other diseases in parts of the country where large numbers of parents don’t vaccinate their kids.

We will apply immunization using the function below. For simplicity, we assume that children are vaccinated exactly on their 5th birthday. We define a vaccination rate ir. On each child’s 5th birthday, we “flip the coin” to determine if they are vaccinated. If so, there disease status is changed to 2. The status of 2 formerly represented recovered. Now it represents immune whether by recovering from the disease or from vaccination. We assume that the vaccine is 100% effective and that immunity is never lost. immunize<-function(pop,ir) { #function to immunize. Assumes that kids are immunized exactly on their five-year-old birthday

#ir is immunization rate

immunize_today<-which(pop[,2]==365*5) #find kids with 5-year-old birthday today nb<-length(immunize_today) #number of kids with 5-year old birthday today

if(nb>0) #if there are any with 5-year birthday today, then check whether immunuized. { pop[immunize_today,]=ifelse(runif(nb)<ir,2,pop[immunize_today,]) #if vaccinated, then change status to immunized } #else leave as is

return(pop)

}

Here is a run with 90% of children immunized at age 5. The model runs for 100 years before the immunization program begins. Because only 5-year-olds are vaccinated, it takes decades before a large portion of the population is immunized. Once this happens, the disease outbreaks are greatly reduced. After the introduction of the vaccine, there are three outbreaks in 200 years. All three are of much smaller magnitude than what was observed before the vaccine was introduced.

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0 50 100 150 200 250 300

0 5

0 0

1 0

0 0

2 0

0 0

90% immunization at age 5

time (years)

n u

m b

e r Susceptible

Infected Recovered/Immune vaccinated

Next, let’s see what happens as the proportion vaccinated is decreased. The next plot shows the same run, but with vaccination rates of 80% rather than 90%. Now we see that the disease outbreaks are larger in magnitude and possibly more frequent. It is easier to see the magnitude of the outbreak in the changes in the number of susceptibles than in the number infected. There are four large outbreaks after the introduction of the vaccine. These are much smaller in magnitude than the pre-vaccine (Where nearly everyone was infected), but still there are large numbers of people getting the disease.

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0 50 100 150 200 250 300

0 5

0 0

1 0

0 0

2 0

0 0

80% immunization at age 5

time (years)

n u

m b

e r Susceptible

Infected Recovered/Immune vaccinated

The next plot shows the case when the vaccination rate is only 50%. Now, the vaccine fails to control the disease. The size of the outbreaks is slightly reduced, but the effect is minimal.

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0 50 100 150 200 250 300

0 5

0 0

1 0

0 0

2 0

0 0

50% immunization at age 5

time (years)

n u

m b

e r Susceptible

Infected Recovered/Immune vaccinated

Herd Immunity

Look back at the first plot. Note that after vaccination has become common that the proportion of Susceptibles and Recovered/Immune was approximately the same as before the epidemic (e.g.the number of susceptible after the vaccine introduction is similar to the average across cycles prior to the introduction). The manner in which people gain immunity changes: before vaccination they gained immunity through getting the disease. After the vaccine, they gain immunity by being vaccinated. However, the numbers are similar.

The key point: Even though the number of susceptibles in the population is about the same as pre-vaccine, the susceptibles mostly don’t catch the disease anymore. This is because the incidence of the disease is much lower in the population. Even though people are susceptible, they don’t get the disease because they don’t encounter infected people. This phenomenon is known as herd immunity

Here is the definition if herd immunity from Wikipedia:

Herd immunity (also called herd effect, community immunity, population immunity, or social immunity) is a form of indirect protection from infectious disease that occurs when a large percentage of a population has become immune to an infection, thereby providing a measure of protection for individuals who are not immune.In a population in which a large number of individuals are immune, chains of infection are likely to be disrupted, which stops or slows the spread of disease. The greater the proportion of individuals in a community who are immune, the smaller the probability that those who are not immune will come into contact with an infectious individual.

Herd immunity is a key part of the benefit of vaccination programs. Because of this, parents choosing to not vaccinate their children has impact beyond their own children. There are many people in the population who cannot be vaccinated. This includes kids that are too young to receive the vaccine and people who with

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compromised immune systems who are unable to be vaccinated. When vaccination rates are high in the rest of the population, then these people are protected by herd immunity. However, when vaccination rates decrease, then herd immunity is lessened and disease outbreaks can occur. We see in the simulations above that the size of disease outbreaks was much higher for 80% vaccination rate than for 90%. That is, even if only 10% of the population decide to not vaccinate their children then many more people will get the disease. Typically, babies and young children are hardest hit because they are too young for vaccination.

Mathematical models have played a large role in epidemiology and particularly in the area of vaccination policy. As we have seen the dynamics are complicated and often non-intuitive. The use of models allows us to explore the ramifications of different vaccination strategies without having to try them out in a real population.

Eradicating Diseases

In order to eradicate a disease. the reproductive rate must be reduced to less than one. Suppose that before the vaccination campaign, that the equilibrium number of susceptibles is S∗. Then, the disease effective intrinsic reproductive rate is

Re = S∗β

γ + µ

If the vaccination rate is v, then the number of susceptibles is reduced by the proportion v and Re becomes

Re = (1 − p)S∗β γ + µ

If Re is reduced below one after the vaccination campaign, then the disease will be eradicated.

Consider the disease simulated in the last three figures. Before the vaccination begins, there are outbreaks when the number of susceptibles reaches about 1800.

The mean infection rate is about 0.05/365 (it varies by age, but is near 0.05 per year) and the recovery rate is 0.1 for all age groups. The background mortality rate is 0.015 except for individuals over 70. Thus, we have

Re = S∗β

γ + µ ≈

1800 ∗ 0.05/365 0.1 + 0.015

= 2.14

Children are vaccinated at 5 years old and are susceptible up until that point. The proportion of children under 5 is about 15%: sum(pop[,2]<5*365)/3000

## [1] 0.1473333

Thus, the total proportion of vaccinated people in the population is

proportion over 5 ∗ vaccination rate

With a 90% vaccination rate for 5-year olds, we have

Re = 2.14 ∗ (1 − 0.9 ∗ 0.85) = 0.5

With an 80% vaccination rate, we have

Re = 2.15 ∗ (1 − 0.8 ∗ 0.85) = 0.42

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With an 50% vaccination rate, we have

Re = 2.15 ∗ (1 − 0.5 ∗ 0.85) = 1.23

Re is below one for 90% and 80% vaccination rates. In both of these cases, the disease was largely eliminated except for small outbreaks coming from new input of infecteds. With a vaccination rate of 50%, Re > 1 and we see that the disease persists in the population with major outbreaks continuing

When do epidemics end?

An epidemic ends when Re drops below one. That is, when infected people on average infect less than one other person.

Recall this expression for Re:

Re = S

N

Nβ

γ + µ =

Sβ

γ + µ

Key points:

1. The effective disease reproductive rate Re is decreased when susceptibles become a lower proportion of the population.

2. The proportion of people who are susceptible drops by people becoming immune.

3. People become immune either by getting the disease or by being vaccinated.

The epidemic ends when the number of people who become immune by getting the disease plus the number of people who become immune by being vaccinated becomes large enough that infected people will not encounter large numbers of susceptible people.

The exact percentage of immune people that is required depends on the details of the disease.

Another key point:

If there is not vaccination (and nothing else changes) then there are likely to be further outbreaks. This is because there will be an inflow of new susceptibles by birth. If they are not vaccinated, then the number of susceptibles will eventually become high enough to fuel a new outbreak. Alternatively, the disease will become endemic: always in the population at a steady low level.

If we don’t want this, then

GET VACCINATED

Age at Infection

The impacts of vaccination are heavily impacted by the age at which infections typically occur. In order to be effective, the vaccine has to be administered at an earlier age than most people will get the disease. With many of the historically deadly diseases (e.g. measles), people would most often catch them in childhood. This is not necessarily because children are more susceptible. If infection rates are high, then most people will catch it within a few years of exposure. For example, suppose that the probability of infection is 20% per year for people of all ages. Then most kids will catch the disease before they are five - not because they are more vulnerable than anyone else to infection, but because it is not possible last longer than a few years without catching it. For a disease with lower infection rates, the average age of infection can be much older.

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Consider the plot below. I set the infection rate to 5% per year for all ages and the recory rate is 10% per day. The bottom plot shows the average age at infection over the population. We see that it varies between 10 and 18 years of age as the intensity of the epidemic varies. When the number of susceptibles is high, then children are catching the disease around 10-12 years old. After an outbreak when the number of suscepibles is low, the average age of infection increases to 16-18. In all cases, it is predominant children who catch the disease, even though the probability of disease transmission is the same for all ages. This is because older people predominantly already caught the disease when they were young and thus are immune.

200 250 300 350 400 450 500

0 1

5 0

0

disease dynamics, beta=0.05/365 for all ages

time (years)

n u

m b

e r

200 250 300 350 400 450 500

1 0

1 4

1 8

Average Age at Infection

a ve

ra g

e a

g e

( ye

a rs

)

Vaccination and Age of Infection

Now, let’s consider what happens when we introduce vaccination:

The first plot below has vaccination at age 5. In this case, the average age at infection drops after the vaccination campaign begins. This is because most infections happen in children who are not vaccinated yet.

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200 300 400 500 600

0 1

5 0

0 vaccinated at age 5

time (years)

n u

m b

e r

200 300 400 500 600

0 1

5 3

0

Average Age at Infection

a ve

ra g

e a

g e

( ye

a rs

)

In the next simulation, children are vaccianted at birth. Before the vaccination campaign, the average age of infection is around six. After the vaccination campaign, the average age of infection increases to the range 14-18. This happens because of herd immunity. Because there are many fewer infected people in the population than before the vaccination, those who are not vaccinated encounter many fewer infected people. Thus, they are typically teenagers when they get the disease.

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200 300 400 500 600

0 1

5 0

0 vaccinated at birth

time (years)

n u

m b

e r

200 300 400 500 600

6 1

2 1

8

Average Age at Infection

a ve

ra g

e a

g e

( ye

a rs

)

Vaccines can strongly impact the age at which people tend to get the disease and frequently will increase the average age at which someone gets the disease

Is this necessarily a good thing? Consider the case of rubella:

Case Study: Rubella

From Wikipedia:

Rubella, also known as German measles or three-day measles, is an infection caused by the rubella virus This disease is often mild with half of people not realizing that they are infected. A rash may start around two weeks after exposure and last for three days. It usually starts on the face and spreads to the rest of the body.The rash is sometimes itchy and is not as bright as that of measles.Swollen lymph nodes are common and may last a few weeks. A fever, sore throat, and fatigue may also occur. In adults joint pain is common. Complications may include bleeding problems, testicular swelling, and inflammation of nerves. Infection during early pregnancy may result in a child born with congenital rubella syndrome (CRS) or miscarriage. Symptoms of CRS include problems with the eyes such as cataracts, ears such as deafness, heart, and brain. Problems are rare after the 20th week of pregnancy. Rubella is usually spread through the air via coughs of people who are infected.People are infectious during the week before and after the appearance of the rash. Babies with CRS may spread the virus for more than a year.[1] Only humans are infected. Insects do not spread the disease. Once recovered, people are immune to future infections. Testing is available that can verify immunity. Diagnosis is confirmed by finding the virus in the blood, throat, or urine. Testing the blood for antibodies may also be useful.[1] Rubella is preventable with the rubella vaccine with a single dose being more than 95Rubella is a common infection in many areas of the world. Each year about 100,000 cases of congenital rubella syndrome occur. Rates of disease have decreased in many areas as a result of vaccination. There are ongoing efforts to eliminate the disease globally. In April 2015 the World Health Organization

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declared the Americas free of rubella transmission. The name "rubella" is from Latin and means little red. It was first described as a separate disease by German physicians in 1814 resulting in the name "German measles."

Children in the US receive the MMR (measles-mumps-rubella) vaccine. The first dose is at around one year old and the second dose is in the range 4-6 years old.

A key characteristic of rubella is that it is usually mild, except when pregnant mothers transmit it to their babies in early pregnancy. The resulting congenital rubella syndrome can be devastating in the newborn baby. Thus, the most important goal in controlling rubella is preventing women of child-bearing age from becoming infected.

Rubella vaccine camapigns commenced in the 1970’s. In a seminal 1983 paper, Anderson and May (Vaccination against rubella and measles: quantitative investigations of different policies) explored the implications of different vaccination policies for rubella.

In the US, the initial rubella vaccination campaign took the strategy of vaccinating all children 15 months or older. The trend over time in cases of rubella and CRS are shown in Figure 4. We see that the number of rubella cases declined after the introduction of the vaccine, as we would expect. However, the number of CRS cases actually increased as the vaccination campaign took hold even as the number of rubella cases dropped very low:

Contrast this to the United Kingdom. There, the vaccination strategy was to vaccinate only girls between ages 10-15. Figure 5 shows the trend in cases of CRS in the UK. We see that the number of cases decreased.

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Why did this happen? It is because of the impact on age at infection of the two policies. The table shown in Figure 6 gives the average age of rubella infection for various countries.

In many countries the average age of infection before vaccination was less than ten years old. This is well before child-bearing age, and so it would not lead to CRS. In the US, the average age was 10-11 before vaccination. After vaccination, the average age of infection increased to 13-16 years. This is approaching the

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lower end of the child-bearing years. Of course, this is average age at infection - some people will get it later. Thus, the US policy greatly reduced the incidence of rubella, but the people who did get it tended to get it later. This was problematic because it meant more pregnant women were getting rubella and transmitting it to their children.

The UK policy targeted only girls shortly before they would reach the age of bearing children. This has the advantage that it doesn’t greatly impact the force of disease infection. That is, there are still many people with rubella in the population and thus those who get rubella will still tend to get it young. Note that that the average age of infection was 9-10 in the UK even after the vaccination campaign.

Note the very young age of infection it the Gambia (2-3 years). Ironically, Gambia had an advantage over more devloped countries: because the disease was so prevalent, nearly everyone caught it very young. Therefore, CRS was very rare.

Anderson and May used mathematical models to study the impact of vaccination strategies on the incidence of CRS. The figure below is from the Anderson and May paper and is based on their mathematical model. The solid lines show the ratio of CRS cases after and before vaccination under the US policy of vaccination versus the proportion p who are vaccinated. The different curves correspond to different values of A, the average age of infection before vaccination. The dashed line shows the same ratio under the UK vaccination policy.

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If the average age of infection is low (which will happen when the disease incidence is high) and if the proportion vaccinated is not high (e.g. 80%) then the US vaccination policy can actually increase the incidence of CRS:

Decreasing the incidence of disease increases the average age of infection. Because rubella is most harmful in pregnant women (by causing CRS), the US vaccination strategy will be counterproductive if 1) The disease prevalence if high before vaccination and 2) the vaccination rate doesn’t approach 100%

In contrast, the UK strategy of vaccinating at age 15 has no downside. Since rubella usually has minimal effect on children, there is not a large cost to vaccinating later. Vaccinating later has the big benefit that it doesn’t change the “force of infection” and therefore the age at which people get rubella. Currently, nearly all children in the US get the MMR (measles-mumps-rubella) vaccine and so the vaccination rate approaches 100% and the policy is effective.

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Many other disease (e.g. measles and chicken pox) are also worse in adults than in children. However, unlike rubella, many such diseases are also harmful in children and so there is a cost to vaccinating later.

The Anderson and May paper had a large impact on understanding of vaccine policy. This is a an example of the usefulness of mathematical models. The dynamics are too complex to work out without the use of models and we don’t want to use real populations as guinea pigs for vaccination strategies. Epidemiological models have played a large role in our understanding of infectious diseases and play a large role in public health policy.

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